💡Okay, so I've got this problem here with three equations involving x, y, and z. Let me see if I can figure it out step by step. First, the equations are:1. y + z = 132. z + x = 143. x + y = 15And I need to find the square root of the product of xyz and the sum of x, y, z. That is, √(xyz(x + y + z)). Hmm, okay.Let me start by trying to find the values of x, y, and z. I have three equations with three variables, so it should be solvable. Maybe I can add all the equations together to find the sum of x, y, and z first.Adding the left sides: (y + z) + (z + x) + (x + y) = 2x + 2y + 2z. That simplifies to 2(x + y + z). Adding the right sides: 13 + 14 + 15. Let me calculate that: 13 + 14 is 27, plus 15 is 42. So, 2(x + y + z) = 42. Dividing both sides by 2 gives x + y + z = 21. Okay, so the sum of x, y, z is 21. That's useful.Now, I need to find individual values of x, y, and z. Let's see. I can use the equations given and substitute the sum we found.From the first equation, y + z = 13. Since x + y + z = 21, subtracting (y + z) from that gives x = 21 - 13 = 8. So, x is 8. That's straightforward.From the second equation, z + x = 14. We know x is 8, so z = 14 - x = 14 - 8 = 6. So, z is 6.From the third equation, x + y = 15. We know x is 8, so y = 15 - x = 15 - 8 = 7. So, y is 7.Let me double-check these values to make sure they satisfy all the original equations.1. y + z = 7 + 6 = 13 ✔️2. z + x = 6 + 8 = 14 ✔️3. x + y = 8 + 7 = 15 ✔️Great, all equations are satisfied. So, x = 8, y = 7, z = 6.Now, I need to compute √(xyz(x + y + z)). Let's break that down.First, compute xyz: 8 * 7 * 6. Let's calculate that step by step.8 * 7 = 56. Then, 56 * 6. 56 * 6 is 336. So, xyz = 336.Next, we already know that x + y + z = 21. So, the product xyz(x + y + z) is 336 * 21.Let me compute that. 336 * 20 is 6720, and 336 * 1 is 336. Adding them together, 6720 + 336 = 7056.So, xyz(x + y + z) = 7056. Now, I need to find the square root of 7056.Hmm, what is √7056? Let me think. I know that 84 squared is 7056 because 80 squared is 6400, and 4 squared is 16. The cross term is 2*80*4=640. So, (80 + 4)^2 = 80^2 + 2*80*4 + 4^2 = 6400 + 640 + 16 = 7056. So, √7056 = 84.Therefore, √(xyz(x + y + z)) = 84.Wait, just to make sure I didn't make any calculation errors. Let me verify the multiplication again.xyz = 8 * 7 * 6. 8*7 is 56, 56*6 is 336. Correct.x + y + z = 21. So, 336 * 21. Let me compute 336 * 21:336 * 20 = 6720336 * 1 = 3366720 + 336 = 7056. Correct.√7056. Let me check 84^2:80^2 = 64004^2 = 16Cross term: 2*80*4 = 640So, (80 + 4)^2 = 6400 + 640 + 16 = 7056. Yes, that's correct.So, everything checks out. The answer is 84.**Final Answer**boxed{84}

💡First, I need to expand both sides of the equation to simplify it. On the left side, multiplying 6 by each term inside the parentheses gives me 72x² + 72x + 66. On the right side, distributing x over (12x - 44) results in 12x² - 44x.Next, I'll move all terms to one side of the equation to set it equal to zero. Subtracting 12x² and adding 44x to both sides, the equation becomes 60x² + 116x + 66 = 0.To simplify further, I'll divide every term by 2, which gives me 30x² + 58x + 33 = 0.Now, I need to factor this quadratic equation. I'll look for two numbers that multiply to 990 (which is 30 multiplied by 33) and add up to 58. The numbers 30 and 33 fit these criteria.Using these numbers, I'll split the middle term: 30x² + 33x + 25x + 33 = 0. Then, I'll factor by grouping:3x(10x + 11) + 3(10x + 11) = 0.This simplifies to (3x + 3)(10x + 11) = 0.Setting each factor equal to zero gives the solutions x = -1 and x = -11/10. Among these, the largest value of x is -1.

💡Okay, so I have this problem about an ice cream shop that's selling two types of ice cream, A and B. They have a total of 50 boxes, and both types cost 15 yuan each to make. The selling price of type A is x yuan, which is at least 20 yuan, and it's an integer. The selling price of type B is always 30 yuan. I need to figure out some algebraic expressions for the profits and then find the maximum total profit, and also determine a donation amount based on some conditions.Starting with part (1), I need to express the profit from selling type A and type B ice creams in terms of x.For type A, when x is 20, they sell 40 boxes. For every 1 yuan increase in x, the sales volume decreases by 2 boxes. So, if x increases by 1, sales decrease by 2. That means the number of boxes sold for type A can be expressed as 40 - 2*(x - 20). Simplifying that, it's 40 - 2x + 40, which is 80 - 2x. Wait, no, that doesn't seem right. Let me check that again.Actually, it's 40 - 2*(x - 20). So, 40 - 2x + 40? Wait, no, that would be 40 - 2*(x - 20) = 40 - 2x + 40? No, that's not correct. Wait, 40 - 2*(x - 20) is 40 - 2x + 40? No, that would be 80 - 2x, but that seems too high because when x is 20, it should be 40. Let me plug in x=20 into 80 - 2x: 80 - 40 = 40, which is correct. So, the number of boxes sold for type A is 80 - 2x.But wait, if x increases beyond 20, say x=21, then 80 - 42 = 38, which is correct because it decreases by 2. So, that seems right.Now, the profit per box for type A is selling price minus cost, which is x - 15. So, the total profit y from type A is (x - 15)*(80 - 2x). Let me expand that: (x - 15)*(80 - 2x) = x*80 - x*2x -15*80 +15*2x = 80x - 2x² - 1200 + 30x = (80x + 30x) - 2x² - 1200 = 110x - 2x² - 1200. So, y = -2x² + 110x - 1200.Okay, that seems correct. Let me double-check: when x=20, y = -2*(400) + 110*20 - 1200 = -800 + 2200 - 1200 = 200. So, profit is 200 yuan when x=20. Since they sold 40 boxes at 20 yuan each, revenue is 800, cost is 40*15=600, so profit is 200. That matches, so y is correct.Now, for type B, the selling price is always 30 yuan. The number of boxes sold for type B would be total boxes minus type A boxes, which is 50 - (80 - 2x) = 50 - 80 + 2x = 2x - 30. So, sales volume for type B is 2x - 30 boxes.Profit per box for type B is 30 - 15 = 15 yuan. So, total profit s from type B is 15*(2x - 30) = 30x - 450.Let me verify that: when x=20, type B sales are 2*20 - 30 = 40 - 30 = 10 boxes. Profit is 10*15=150 yuan. Total profit y + s = 200 + 150 = 350 yuan. Let me check if that makes sense. Total revenue is 40*20 + 10*30 = 800 + 300 = 1100. Total cost is 50*15=750. So, profit is 1100 - 750 = 350. Correct.So, part (1) is done. Now, part (2): total profit w is y + s. So, w = (-2x² + 110x - 1200) + (30x - 450) = -2x² + 140x - 1650.To find the maximum value of w, since it's a quadratic function with a negative coefficient on x², the maximum occurs at the vertex. The x-coordinate of the vertex is at -b/(2a) = -140/(2*(-2)) = -140/(-4) = 35. So, x=35.But x must be an integer and at least 20. Let me check if x=35 is within the valid range. Since the sales volume for type A is 80 - 2x, when x=35, it's 80 - 70 = 10 boxes. Type B would be 2*35 - 30 = 70 - 30 = 40 boxes. Both are positive integers, so that's valid.Now, calculate the maximum profit w at x=35: w = -2*(35)^2 + 140*35 - 1650. Let's compute that step by step.35 squared is 1225. So, -2*1225 = -2450. 140*35: 140*30=4200, 140*5=700, so total 4900. So, -2450 + 4900 = 2450. Then, 2450 - 1650 = 800. So, maximum profit is 800 yuan when x=35.Wait, let me verify that with actual sales. At x=35, type A sells 10 boxes, so revenue is 10*35=350. Type B sells 40 boxes, revenue is 40*30=1200. Total revenue is 350 + 1200=1550. Total cost is 50*15=750. Profit is 1550 - 750=800. Correct.Now, part (3): For each box of type B sold, the shop donates a yuan to the Hope Project. So, the total donation is a*(number of type B boxes sold) = a*(2x - 30). So, the new total profit w' = w - a*(2x - 30). We are told that the maximum value of w' is 722 yuan. We need to find a.So, w' = (-2x² + 140x - 1650) - a*(2x - 30). Let's expand that: w' = -2x² + 140x - 1650 - 2a x + 30a. Combine like terms: w' = -2x² + (140 - 2a)x + (-1650 + 30a).We need to find the maximum of this quadratic function and set it equal to 722. The maximum occurs at x = -b/(2a) where the quadratic is ax² + bx + c. Here, a=-2, b=(140 - 2a). So, x = -(140 - 2a)/(2*(-2)) = (140 - 2a)/4 = (70 - a)/2.But x must be an integer between 20 and 40, and the sales volumes must be positive integers. So, (70 - a)/2 must be an integer, or close to it. Let me denote x* = (70 - a)/2. Since x must be an integer, x* must be either floor or ceiling of (70 - a)/2.But since we are looking for the maximum, we can assume that x* is an integer. So, (70 - a) must be even, meaning a must be even because 70 is even. So, a is even.Now, the maximum value of w' is 722. So, we need to compute w' at x = x* and set it equal to 722.But this might be complicated because a is involved in both the x* and the expression for w'. Alternatively, we can take the derivative or use calculus, but since it's a quadratic, we can use the vertex formula.The maximum value of w' is given by c - b²/(4a). Wait, no, the maximum value is at x = -b/(2a), which we have as x* = (70 - a)/2.So, substituting x* into w', we get:w' = -2*(x*)² + (140 - 2a)*x* + (-1650 + 30a).But x* = (70 - a)/2, so let's substitute:w' = -2*((70 - a)/2)^2 + (140 - 2a)*((70 - a)/2) + (-1650 + 30a).Let me compute each term step by step.First term: -2*((70 - a)/2)^2 = -2*( (70 - a)^2 )/4 = - (70 - a)^2 / 2.Second term: (140 - 2a)*((70 - a)/2) = (140 - 2a)(70 - a)/2.Third term: -1650 + 30a.So, w' = - (70 - a)^2 / 2 + (140 - 2a)(70 - a)/2 + (-1650 + 30a).Let me compute each part:First term: - (70 - a)^2 / 2.Second term: (140 - 2a)(70 - a)/2.Let me expand (140 - 2a)(70 - a):= 140*70 - 140a - 2a*70 + 2a²= 9800 - 140a - 140a + 2a²= 9800 - 280a + 2a².So, second term is (9800 - 280a + 2a²)/2 = 4900 - 140a + a².Third term: -1650 + 30a.Now, combining all terms:w' = - (70 - a)^2 / 2 + 4900 - 140a + a² - 1650 + 30a.Simplify term by term:First term: - (4900 - 140a + a²)/2 = -2450 + 70a - 0.5a².Second term: 4900 - 140a + a².Third term: -1650 + 30a.Now, combine all together:(-2450 + 70a - 0.5a²) + (4900 - 140a + a²) + (-1650 + 30a).Combine like terms:Constants: -2450 + 4900 - 1650 = (-2450 - 1650) + 4900 = (-4100) + 4900 = 800.a terms: 70a - 140a + 30a = (-70a) + 30a = -40a.a² terms: -0.5a² + a² = 0.5a².So, w' = 0.5a² - 40a + 800.We are told that the maximum value of w' is 722. So, 0.5a² - 40a + 800 = 722.Subtract 722 from both sides: 0.5a² - 40a + 78 = 0.Multiply both sides by 2 to eliminate the decimal: a² - 80a + 156 = 0.Now, solve for a using quadratic formula: a = [80 ± sqrt(80² - 4*1*156)] / 2.Compute discriminant: 6400 - 624 = 5776.sqrt(5776) = 76.So, a = [80 ±76]/2.Thus, a = (80 +76)/2 = 156/2 =78, or a=(80 -76)/2=4/2=2.But a must be between 0 and 10, so a=2.Let me verify that. If a=2, then the total profit w' =0.5*(4) -40*2 +800= 2 -80 +800=722. Correct.So, the value of a is 2.

💡Okay, so I've got these two logarithm problems to solve. Let me take them one at a time. Starting with the first one: Calculate (log_2)^2 + (log_{20} + 2)log_5 + log_4. Hmm, that looks a bit complicated, but maybe I can break it down. First, I need to understand what each term means. The first term is (log_2)^2. Wait, is that the square of the logarithm base 2 of something? But there's no number specified. Maybe it's a typo? Or perhaps it's supposed to be log_2 of something else? Hmm, maybe I should assume it's log_2 of 10 or something? Wait, no, that doesn't make much sense. Maybe it's log_2 2? Because if it's just log_2, that's not a complete expression. Wait, let me think again. The problem is written as (log_2)^2 + (log_{20} + 2)log_5 + log_4. Maybe it's supposed to be log_2 of 10 squared? Or is it the square of log_2? That doesn't quite make sense. Maybe it's log_2 2 squared? Because log_2 2 is 1, so 1 squared is still 1. But then the next term is (log_{20} + 2)log_5. Again, similar issue. Is that log_{20} of something? Or is it log_{20} 20? Because log_{20} 20 is 1, so 1 + 2 is 3, and then multiplied by log_5 of something. Hmm, this is confusing. Maybe I need to interpret it differently. Wait, perhaps the notation is that the logarithm is applied to 10? So, like log_2 10, log_{20} 10, and log_5 10, and log_4 10? That might make sense because in some contexts, if the argument is omitted, it's assumed to be 10. So maybe the expression is (log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10. If that's the case, then I can work with that. Let me write it out:(log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10.Okay, now I can try to compute each term separately.First term: (log_2 10)^2. I know that log_2 10 is approximately 3.3219, but maybe I can express it in terms of other logarithms. Alternatively, I can use the change of base formula: log_b a = frac{ln a}{ln b}. So, log_2 10 = frac{ln 10}{ln 2}. Similarly, log_{20} 10 = frac{ln 10}{ln 20}, and log_5 10 = frac{ln 10}{ln 5}, and log_4 10 = frac{ln 10}{ln 4}. But maybe instead of calculating numerically, I can find a way to simplify the expression algebraically. Let me see.Let me denote x = log_2 10, y = log_5 10, and z = log_{20} 10, and w = log_4 10. Then the expression becomes x^2 + (z + 2)y + w. But I need to relate these variables. Maybe I can express all logarithms in terms of a common base, say base 10. Wait, but if I do that, I can use the change of base formula:log_b a = frac{log a}{log b}.So, for example, x = log_2 10 = frac{log 10}{log 2} = frac{1}{log 2}, since log 10 = 1 in base 10. Similarly, y = log_5 10 = frac{1}{log 5}, z = log_{20} 10 = frac{1}{log 20}, and w = log_4 10 = frac{1}{log 4}. So, substituting back, the expression becomes:left(frac{1}{log 2}right)^2 + left(frac{1}{log 20} + 2right)left(frac{1}{log 5}right) + frac{1}{log 4}.Hmm, that seems a bit messy, but maybe I can find some relationships between the logarithms. I know that log 20 = log (4 times 5) = log 4 + log 5. Similarly, log 4 = 2 log 2. So, let's write that down:log 20 = log 4 + log 5 = 2 log 2 + log 5.So, frac{1}{log 20} = frac{1}{2 log 2 + log 5}.Similarly, frac{1}{log 4} = frac{1}{2 log 2}.So, substituting back into the expression:left(frac{1}{log 2}right)^2 + left(frac{1}{2 log 2 + log 5} + 2right)left(frac{1}{log 5}right) + frac{1}{2 log 2}.This is getting complicated, but maybe I can find a way to combine terms. Let me denote a = log 2 and b = log 5 for simplicity. Then, the expression becomes:left(frac{1}{a}right)^2 + left(frac{1}{2a + b} + 2right)left(frac{1}{b}right) + frac{1}{2a}.Simplify each term:First term: frac{1}{a^2}.Second term: left(frac{1}{2a + b} + 2right)left(frac{1}{b}right) = frac{1}{b(2a + b)} + frac{2}{b}.Third term: frac{1}{2a}.So, putting it all together:frac{1}{a^2} + frac{1}{b(2a + b)} + frac{2}{b} + frac{1}{2a}.Hmm, maybe I can combine the terms with frac{1}{b}:frac{1}{a^2} + frac{1}{b(2a + b)} + frac{2}{b} + frac{1}{2a} = frac{1}{a^2} + frac{1}{b(2a + b)} + frac{2}{b} + frac{1}{2a}.Let me see if I can find a common denominator or something. Alternatively, maybe I can express everything in terms of a and b and see if it simplifies.Wait, I also know that a + b = log 2 + log 5 = log 10 = 1. So, a + b = 1. That might be useful.So, a + b = 1 implies that b = 1 - a. Let me substitute b = 1 - a into the expression.First term: frac{1}{a^2}.Second term: frac{1}{(1 - a)(2a + (1 - a))} = frac{1}{(1 - a)(a + 1)} = frac{1}{(1 - a)(1 + a)} = frac{1}{1 - a^2}.Third term: frac{2}{1 - a}.Fourth term: frac{1}{2a}.So, the expression becomes:frac{1}{a^2} + frac{1}{1 - a^2} + frac{2}{1 - a} + frac{1}{2a}.Hmm, maybe I can combine the first two terms:frac{1}{a^2} + frac{1}{1 - a^2} = frac{1 - a^2 + a^2}{a^2(1 - a^2)} = frac{1}{a^2(1 - a^2)}.So, now the expression is:frac{1}{a^2(1 - a^2)} + frac{2}{1 - a} + frac{1}{2a}.Hmm, not sure if that helps. Maybe I can find a common denominator for all terms. Let's see.The denominators are a^2(1 - a^2), 1 - a, and 2a. The common denominator would be 2a^2(1 - a^2). Let's rewrite each term:First term: frac{1}{a^2(1 - a^2)} = frac{2}{2a^2(1 - a^2)}.Second term: frac{2}{1 - a} = frac{2 times 2a^2(1 + a)}{2a^2(1 - a^2)} = frac{4a^2(1 + a)}{2a^2(1 - a^2)}.Wait, that seems complicated. Alternatively, maybe I can express frac{2}{1 - a} as frac{2(1 + a)}{(1 - a)(1 + a)} = frac{2(1 + a)}{1 - a^2}. Then, the second term becomes frac{2(1 + a)}{1 - a^2}.Similarly, the third term is frac{1}{2a} = frac{(1 - a^2)}{2a(1 - a^2)}.So, putting it all together:frac{1}{a^2(1 - a^2)} + frac{2(1 + a)}{1 - a^2} + frac{1 - a^2}{2a(1 - a^2)}.Hmm, this is getting too convoluted. Maybe I need a different approach.Wait, going back to the original expression: (log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10.Maybe instead of trying to express everything in terms of a and b, I can use properties of logarithms to simplify the expression.I know that log_{20} 10 can be expressed as frac{log_2 10}{log_2 20}. Similarly, log_5 10 = frac{log_2 10}{log_2 5}. Wait, let me try that.Let me denote x = log_2 10, so log_5 10 = frac{x}{log_2 5} and log_{20} 10 = frac{x}{log_2 20}.So, substituting back into the expression:x^2 + left(frac{x}{log_2 20} + 2right)left(frac{x}{log_2 5}right) + log_4 10.Hmm, but log_4 10 = frac{log_2 10}{log_2 4} = frac{x}{2}, since log_2 4 = 2.So, the expression becomes:x^2 + left(frac{x}{log_2 20} + 2right)left(frac{x}{log_2 5}right) + frac{x}{2}.Now, let's compute log_2 20 and log_2 5.log_2 20 = log_2 (4 times 5) = log_2 4 + log_2 5 = 2 + log_2 5.Let me denote c = log_2 5, so log_2 20 = 2 + c.So, substituting back:x^2 + left(frac{x}{2 + c} + 2right)left(frac{x}{c}right) + frac{x}{2}.Now, let's expand the second term:left(frac{x}{2 + c} + 2right)left(frac{x}{c}right) = frac{x^2}{c(2 + c)} + frac{2x}{c}.So, the entire expression becomes:x^2 + frac{x^2}{c(2 + c)} + frac{2x}{c} + frac{x}{2}.Hmm, maybe I can factor out x^2 and x:x^2 left(1 + frac{1}{c(2 + c)}right) + x left(frac{2}{c} + frac{1}{2}right).This still looks complicated. Maybe I need to find a relationship between x and c.Since x = log_2 10 and c = log_2 5, we can note that x = log_2 (2 times 5) = log_2 2 + log_2 5 = 1 + c. So, x = 1 + c.Therefore, c = x - 1.Substituting back into the expression:x^2 left(1 + frac{1}{(x - 1)(2 + x - 1)}right) + x left(frac{2}{x - 1} + frac{1}{2}right).Simplify the denominators:2 + x - 1 = x + 1.So, the expression becomes:x^2 left(1 + frac{1}{(x - 1)(x + 1)}right) + x left(frac{2}{x - 1} + frac{1}{2}right).Simplify the first part inside the brackets:1 + frac{1}{(x - 1)(x + 1)} = 1 + frac{1}{x^2 - 1}.So, the expression is:x^2 left(1 + frac{1}{x^2 - 1}right) + x left(frac{2}{x - 1} + frac{1}{2}right).Let me compute each part separately.First part: x^2 left(1 + frac{1}{x^2 - 1}right) = x^2 + frac{x^2}{x^2 - 1}.Second part: x left(frac{2}{x - 1} + frac{1}{2}right) = frac{2x}{x - 1} + frac{x}{2}.So, the entire expression is:x^2 + frac{x^2}{x^2 - 1} + frac{2x}{x - 1} + frac{x}{2}.Hmm, this is still quite involved. Maybe I can combine the first two terms:x^2 + frac{x^2}{x^2 - 1} = x^2 left(1 + frac{1}{x^2 - 1}right) = x^2 left(frac{x^2 - 1 + 1}{x^2 - 1}right) = x^2 left(frac{x^2}{x^2 - 1}right) = frac{x^4}{x^2 - 1}.So, now the expression is:frac{x^4}{x^2 - 1} + frac{2x}{x - 1} + frac{x}{2}.Hmm, maybe I can factor x^4 as (x^2)^2 and see if that helps. Alternatively, perhaps I can express frac{x^4}{x^2 - 1} as x^2 + frac{x^2}{x^2 - 1}, but that just brings us back.Wait, another approach: since x = log_2 10, and we know that 10 = 2 times 5, so x = 1 + log_2 5 = 1 + c, as before. But I'm not sure if that helps here.Alternatively, maybe I can assign a numerical value to x to see if the expression simplifies. Let's compute x = log_2 10 approx 3.321928.Then, let's compute each term:First term: x^2 approx (3.321928)^2 approx 11.032.Second term: frac{x^2}{x^2 - 1} approx frac{11.032}{11.032 - 1} approx frac{11.032}{10.032} approx 1.099.Third term: frac{2x}{x - 1} approx frac{2 times 3.321928}{3.321928 - 1} approx frac{6.643856}{2.321928} approx 2.861.Fourth term: frac{x}{2} approx frac{3.321928}{2} approx 1.660964.Adding them all up: 11.032 + 1.099 + 2.861 + 1.660964 ≈ 16.652.Wait, that's a number, but I was expecting maybe an integer or a simpler expression. Maybe I made a mistake in interpreting the original problem.Wait, going back to the original problem: Calculate (log_2)^2 + (log_{20} + 2)log_5 + log_4.I assumed that each logarithm was base 10, but maybe that's not the case. Maybe the logarithms are in different bases, but the arguments are missing. Alternatively, perhaps the expression is supposed to be in terms of logarithms with the same base.Wait, another interpretation: Maybe the expression is (log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10, as I thought before. But when I calculated it numerically, I got approximately 16.652, which doesn't seem like a nice number. Maybe I made a mistake in the calculation.Alternatively, perhaps the expression simplifies to 3, as the assistant suggested. Let me check that.Wait, the assistant's solution was:(log_2)^2 + (log_{20} + 2)log_5 + log_4 = log_2(log_2 + log_5) + log_5 + 2(log_2 + log_5) = log_2 + log_5 + 2 = 3.Wait, that seems too simplistic. How did they get from (log_2)^2 + (log_{20} + 2)log_5 + log_4 to log_2(log_2 + log_5) + log_5 + 2(log_2 + log_5)?Let me see. They seem to have distributed the terms:- (log_2)^2 is written as log_2 cdot log_2.- (log_{20} + 2)log_5 is written as log_{20} cdot log_5 + 2 cdot log_5.- log_4 is just log_4.Then, they combined terms:log_2 cdot log_2 + log_{20} cdot log_5 + 2 cdot log_5 + log_4.But then they factor out log_2 from the first two terms:log_2(log_2 + log_5) + log_5 + 2(log_2 + log_5).Wait, but how does log_{20} cdot log_5 become log_2 cdot log_5? That doesn't seem right unless log_{20} = log_2, which isn't true.Wait, maybe they used a change of base formula. Let me see.log_{20} = frac{log_2}{log_2 20}, but that's not helpful. Alternatively, log_{20} 10 = frac{log_2 10}{log_2 20}.Wait, perhaps they assumed that all logarithms are base 10, so log_{20} = log_{20} 10, and similarly for others.But in that case, the expression would be:(log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10.Which is what I was working on earlier, but the assistant's solution seems to have a different approach.Wait, maybe they used the property that log_a b + log_b a = frac{1}{log_b a} + log_b a, but I'm not sure.Alternatively, perhaps they made a mistake in their solution, as my numerical calculation didn't match their result.Wait, let me try to compute the expression as per the assistant's solution.They wrote:(log_2)^2 + (log_{20} + 2)log_5 + log_4 = log_2(log_2 + log_5) + log_5 + 2(log_2 + log_5) = log_2 + log_5 + 2 = 3.But this seems to assume that (log_2)^2 = log_2 cdot log_2, which is true, but then they have (log_{20} + 2)log_5 = log_{20} cdot log_5 + 2 cdot log_5, which is also true. Then, they factor out log_2 from the first two terms, but that would require log_{20} = log_2, which isn't correct.Wait, unless they are using a different interpretation of the logarithms. Maybe they are assuming that the logarithms are in the same base, say base 10, and that the arguments are 10, but then the expression would be as I wrote earlier.Alternatively, maybe they are using the fact that log_{20} = log_{2 times 10} = log_2 + log_{10}, but that's not correct because log_{20} a = frac{log a}{log 20}, not log_2 a + log_{10} a.Wait, perhaps they made a mistake in their solution. Let me try to see if the expression can indeed simplify to 3.Wait, if I take the expression as (log_2 10)^2 + (log_{20} 10 + 2)log_5 10 + log_4 10, and plug in the approximate values:log_2 10 approx 3.3219, so (log_2 10)^2 approx 11.032.log_{20} 10 approx 0.5306, so log_{20} 10 + 2 approx 2.5306.log_5 10 approx 1.4307, so (log_{20} 10 + 2)log_5 10 approx 2.5306 times 1.4307 approx 3.614.log_4 10 approx 1.660964.Adding them up: 11.032 + 3.614 + 1.660964 ≈ 16.307, which is nowhere near 3.So, the assistant's solution must be incorrect, or perhaps I'm misinterpreting the original problem.Wait, maybe the original problem was written differently. Let me check again:(1) Calculate (log_2)^2 + (log_{20} + 2)log_5 + log_4;(2) Given that log_5 3 = a and log_5 4 = b, represent log_{25} 144 in terms of a and b.Hmm, perhaps the first problem is written without specifying the arguments, which is confusing. Maybe it's supposed to be log_2 10, log_{20} 10, etc., but without the arguments, it's unclear.Alternatively, maybe the expression is in terms of logarithms with the same base, say base 10, but without the arguments, it's ambiguous.Alternatively, perhaps the expression is written in a different notation, where log_2 is the logarithm of 2, not base 2. That is, log_2 could mean log(2), where the base is 10. Similarly, log_{20} would be log(20), etc.If that's the case, then the expression is:(log 2)^2 + (log 20 + 2)log 5 + log 4.Let me compute that.First, compute each logarithm:log 2 approx 0.3010, so (log 2)^2 approx 0.0906.log 20 approx 1.3010, so log 20 + 2 approx 3.3010.log 5 approx 0.6990, so (log 20 + 2)log 5 approx 3.3010 times 0.6990 approx 2.310.log 4 approx 0.6020.Adding them up: 0.0906 + 2.310 + 0.6020 ≈ 3.0026, which is approximately 3. So, the answer is 3.Ah, so that makes sense. The original problem was written without specifying the base, but if we assume that all logarithms are base 10, and the arguments are the numbers written after the log, then the expression simplifies to approximately 3.So, the correct interpretation is:(log 2)^2 + (log 20 + 2)log 5 + log 4.Which, when computed, gives approximately 3.Therefore, the first answer is 3.Now, moving on to the second problem:Given that log_5 3 = a and log_5 4 = b, represent log_{25} 144 in terms of a and b.Okay, so we need to express log_{25} 144 using a and b, where a = log_5 3 and b = log_5 4.First, let's recall that log_{25} 144 can be rewritten using the change of base formula. Since 25 is 5^2, we can write:log_{25} 144 = frac{log_5 144}{log_5 25}.We know that log_5 25 = log_5 5^2 = 2.So, log_{25} 144 = frac{log_5 144}{2}.Now, we need to express log_5 144 in terms of a and b.Note that 144 is 12 squared, and 12 is 3 times 4. So, 144 = (3 times 4)^2 = 3^2 times 4^2.Therefore, log_5 144 = log_5 (3^2 times 4^2) = log_5 3^2 + log_5 4^2 = 2 log_5 3 + 2 log_5 4.Given that log_5 3 = a and log_5 4 = b, this becomes:log_5 144 = 2a + 2b = 2(a + b).Therefore, log_{25} 144 = frac{2(a + b)}{2} = a + b.So, the answer is a + b.Wait, let me double-check that.log_{25} 144 = frac{log_5 144}{log_5 25} = frac{log_5 (12^2)}{2} = frac{2 log_5 12}{2} = log_5 12.But log_5 12 = log_5 (3 times 4) = log_5 3 + log_5 4 = a + b.Yes, that's correct. So, log_{25} 144 = a + b.Therefore, the second answer is a + b.

💡Okay, so I have this system of equations to solve:2x + 5y = 26 ...(1)4x - 2y = 4 ...(2)Hmm, I remember that there are a couple of ways to solve systems like this. One is substitution, and the other is elimination. I think elimination might be easier here because the coefficients of x in both equations are multiples of 2. Let me see.First, maybe I can make the coefficients of x the same in both equations so that I can subtract one equation from the other and eliminate x. To do that, I can multiply equation (1) by 2. Let's try that.Multiplying equation (1) by 2:2*(2x + 5y) = 2*26Which gives:4x + 10y = 52 ...(3)Now, equation (2) is:4x - 2y = 4 ...(2)If I subtract equation (2) from equation (3), the 4x terms will cancel out. Let's do that.(4x + 10y) - (4x - 2y) = 52 - 4Simplifying the left side:4x + 10y - 4x + 2y = 12yAnd the right side:52 - 4 = 48So, 12y = 48Now, to find y, I divide both sides by 12:y = 48 / 12y = 4Okay, so y is 4. Now, I need to find x. I can plug y = 4 back into either equation (1) or (2). Let's use equation (1) because the numbers seem smaller.Plugging y = 4 into equation (1):2x + 5*4 = 262x + 20 = 26Subtract 20 from both sides:2x = 6Divide both sides by 2:x = 3So, x is 3 and y is 4. Let me check if these values satisfy both equations to make sure I didn't make a mistake.Checking equation (1):2*3 + 5*4 = 6 + 20 = 26Yes, that's correct.Checking equation (2):4*3 - 2*4 = 12 - 8 = 4That also works. Okay, so I think I did it right. The solution is x = 3 and y = 4.

💡Okay, so I have this problem about a circle given by the equation ( x^2 + y^2 + Dx + Ey + F = 0 ). It says that the circle is tangent to the x-axis at the origin, and I need to figure out which of the options A, B, C, or D is correct based on the values of D, E, and F.First, I remember that the general equation of a circle can be rewritten in the standard form by completing the squares. The standard form is ( (x - h)^2 + (y - k)^2 = r^2 ), where (h, k) is the center and r is the radius.So, let me try to rewrite the given equation into this standard form. Starting with ( x^2 + y^2 + Dx + Ey + F = 0 ), I can group the x terms and y terms together:( x^2 + Dx + y^2 + Ey = -F ).Now, I need to complete the square for both x and y.For the x terms: ( x^2 + Dx ). To complete the square, I take half of D, which is ( D/2 ), and square it, getting ( (D/2)^2 = D^2/4 ). So, I add and subtract this inside the equation:( (x^2 + Dx + D^2/4) - D^2/4 ).Similarly, for the y terms: ( y^2 + Ey ). Half of E is ( E/2 ), and squaring it gives ( E^2/4 ). So, I add and subtract this as well:( (y^2 + Ey + E^2/4) - E^2/4 ).Putting it all together, the equation becomes:( (x + D/2)^2 - D^2/4 + (y + E/2)^2 - E^2/4 = -F ).Simplifying the constants on the left side:( (x + D/2)^2 + (y + E/2)^2 = D^2/4 + E^2/4 - F ).So, the standard form is:( (x + D/2)^2 + (y + E/2)^2 = (D^2 + E^2)/4 - F ).From this, I can see that the center of the circle is at ( (-D/2, -E/2) ) and the radius is ( sqrt{(D^2 + E^2)/4 - F} ).Now, the problem states that the circle is tangent to the x-axis at the origin. Let me think about what that means.First, being tangent to the x-axis implies that the circle touches the x-axis at exactly one point. Since it's tangent at the origin, that point of tangency is (0, 0). So, the origin lies on the circle.Also, the radius at the point of tangency should be perpendicular to the tangent line. Since the tangent line is the x-axis, which is horizontal, the radius at the origin must be vertical. That means the center of the circle must lie directly above or below the origin on the y-axis.Wait, if the radius is vertical, then the x-coordinate of the center must be 0. Because if the center is at (h, k), and the radius is vertical, then h must be 0. So, the center is at (0, k).Looking back at the center coordinates from the standard form, which are ( (-D/2, -E/2) ). So, for the x-coordinate to be 0, we have:( -D/2 = 0 ) => ( D = 0 ).So, D must be 0.Now, since the center is at (0, -E/2), and the circle is tangent to the x-axis at the origin, the distance from the center to the x-axis must be equal to the radius.The distance from the center (0, -E/2) to the x-axis is the absolute value of the y-coordinate, which is | -E/2 | = |E| / 2.This distance must equal the radius. So, let's find the radius.From the standard form, the radius squared is ( (D^2 + E^2)/4 - F ). Since D is 0, this simplifies to ( (0 + E^2)/4 - F = E^2/4 - F ).Therefore, the radius is ( sqrt{E^2/4 - F} ).But we also know that the radius must equal |E| / 2 because the distance from the center to the x-axis is |E| / 2.So, setting them equal:( sqrt{E^2/4 - F} = |E| / 2 ).Let me square both sides to eliminate the square root:( E^2/4 - F = (|E| / 2)^2 ).But ( (|E| / 2)^2 = E^2 / 4 ), so:( E^2/4 - F = E^2/4 ).Subtracting ( E^2/4 ) from both sides:( -F = 0 ) => ( F = 0 ).Wait, but hold on. If F is 0, then the equation of the circle becomes ( x^2 + y^2 + Ey = 0 ). But the circle is supposed to be tangent to the x-axis at the origin. Let me check if F can be 0.If F is 0, then plugging in the origin (0, 0) into the equation gives 0 + 0 + 0 + 0 + 0 = 0, which satisfies the equation. So, the origin is on the circle.But also, the radius is |E| / 2, and the center is at (0, -E/2). So, if E is not zero, the center is at (0, -E/2), and the radius is |E| / 2. So, the circle would extend from y = -E/2 - |E|/2 to y = -E/2 + |E|/2.But if E is positive, then the center is at (0, -E/2), and the radius is E/2, so the circle would extend from y = -E/2 - E/2 = -E to y = -E/2 + E/2 = 0. So, it touches the x-axis at y=0, which is the origin.Similarly, if E is negative, say E = -k where k is positive, then the center is at (0, k/2), and the radius is |E| / 2 = k/2. So, the circle would extend from y = k/2 - k/2 = 0 to y = k/2 + k/2 = k. So, again, it touches the x-axis at y=0, the origin.So, F must be 0? But wait, in my earlier step, I concluded that F = 0. But looking back at the options, option B is E = F = 0, D ≠ 0. But earlier, I found that D must be 0. So, if D is 0, F is 0, then E must be non-zero because otherwise, the circle would collapse to a point.Wait, if E is 0 as well, then the equation becomes ( x^2 + y^2 = 0 ), which is just the origin. But that's not a circle; it's a single point. So, E cannot be zero. Therefore, E ≠ 0, D = 0, and F = 0.Looking back at the options:A: D=0, E≠0, F≠0;B: E=F=0, D≠0;C: D=F=0, E≠0;D: D=E=0, F≠0.So, according to my conclusion, D=0, E≠0, F=0. But none of the options exactly say that. Option C says D=F=0, E≠0. That matches my conclusion.Wait, but earlier I thought F must be 0, but in the standard form, the radius squared is ( E^2/4 - F ). If F is 0, then the radius is ( E/2 ), which is correct. So, F must be 0.But in the options, option C is D=F=0, E≠0, which is exactly what I have. So, why did I initially think the answer was A?Wait, maybe I made a mistake in my earlier reasoning. Let me go back.I concluded that F = 0 because ( sqrt{E^2/4 - F} = |E| / 2 ) led to F = 0. But let me double-check that.Starting from:( sqrt{E^2/4 - F} = |E| / 2 ).Squaring both sides:( E^2/4 - F = E^2/4 ).Subtracting ( E^2/4 ) from both sides:( -F = 0 ) => ( F = 0 ).Yes, that seems correct. So, F must be 0.But in the options, option C is D=F=0, E≠0, which is correct. So, why did I initially think the answer was A?Wait, maybe I confused something else. Let me check the options again.Option A: D=0, E≠0, F≠0.Option B: E=F=0, D≠0.Option C: D=F=0, E≠0.Option D: D=E=0, F≠0.So, according to my conclusion, it's option C.But in the initial analysis, the assistant said the answer was A, but that must be incorrect because F must be 0.Wait, perhaps the assistant made a mistake. Let me see.In the initial analysis, the assistant said:"Since the circle is tangent to the x-axis at the origin, the center of the circle is on the y-axis, which means D=0. The absolute value of the y-coordinate of the center equals the radius, and we have F≠0, E≠0. Therefore, the correct answer is A."But according to my detailed reasoning, F must be 0. So, the assistant's initial answer was incorrect.Therefore, the correct answer should be option C: D=F=0, E≠0.Wait, but let me double-check.If the circle is tangent to the x-axis at the origin, then the origin is on the circle, so plugging (0,0) into the equation gives 0 + 0 + 0 + 0 + F = 0 => F = 0.So, F must be 0. Therefore, option C is correct.But in the initial analysis, the assistant said F≠0, which is incorrect.So, the correct answer is option C.